A1-homotopy theory of noncommutative motives

In this article we continue the development of a theory of noncommutative motives. We construct categories of A1-homotopy noncommutative motives, described their universal properties, and compute their spectra of morphisms in terms of Karoubi-Villamayor's K-theory (KV) and Weibel's homotopy K-theory (KH). As an application, we obtain a complete classification of all the natural transformations defined on KV, KH. This leads to a streamlined construction of Weibel's homotopy Chern character from KV to periodic cyclic homology. Along the way we extend Dwyer-Friedlander's etale K-theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.